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I have a question about the definition of Cauchy's Integral Formula:

Let $\gamma$ be a simpled closed positively oriented contour. If f is analytic in some simply connected domain $D$ containing $\gamma$ and $z_0$ is any point inside $\gamma$ then: $$ f(z_0) = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{z-z_0} dz $$ Fundamentals of Complex Analysis (Pearson Education 2014)

My question is regarding the choice of contour above, is this not also true if $\gamma$ is a loop (see example image below) or why do we have to add that the contour needs to also be simple? The proof of the formula involves using the Deformation Invariance Theorem (deformation of contours) to deform the contour $\gamma$ in question to a circle around the point $z_0$ and we should still be able to do this step if the contour in question is a loop. Am I missing something obvious or are there maybe applications after complex analysis that constrains the formula, which the author might not notice the reader about in my textbook.

enter image description here

Edits:

  • Missed the $2\pi i$ part in the formula.
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    $\begingroup$ If it is not simple, what is the meaning of “positively oriented” then? $\endgroup$ Commented Jan 19, 2022 at 21:41
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    $\begingroup$ Some versions of the formula have an additional accounting for the index of the loop around $z_0$. $\endgroup$
    – copper.hat
    Commented Jan 19, 2022 at 21:43
  • $\begingroup$ We can still have a closed contour that is positively oriented and if it was negatively orientated we would just have to change the sign of the integral and everything would work out just fine. please let me know if I'm missing something but is the orientation crucial for the formula and the distinction between "simple closed contours" and "closed contours" $\endgroup$ Commented Jan 19, 2022 at 21:49
  • $\begingroup$ @copper.hat could you link me to any resources that go into details about that, would love to read up! thanks for the quick answer! $\endgroup$ Commented Jan 19, 2022 at 21:52
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    $\begingroup$ @JamesBlond Theorem 5.4 in Conway's "Functions of one complex variable", 2nd Ed. Basically $\eta(\gamma; a)f(a) = {1 \over 2 \pi i} \int_\gamma {f(z) \over z-a} dz $. $\endgroup$
    – copper.hat
    Commented Jan 19, 2022 at 22:01

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First of all, notice that you missed a $2 \pi i$ factor in the formula.

Now to your question. The thing is, that if you integrate in such a loop, you can "divide" that loop into two closed curves and by additive properties of the integrals, write the integral as the sum of two other integrals. however, if one of those divisions of the loops doesn't include any singularities, then that integral reduces to $0$ (see Cauchy-Goursat Theorem).

So, in the end, it does not matter if $\gamma$ is a loop or just a simple-closed curve, because a loop could always be treated as a union of closed curves.

I hope my answer helps!

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  • $\begingroup$ Thank you Luciano, I edited the post to make it right now. $\endgroup$ Commented Jan 19, 2022 at 21:53
  • $\begingroup$ Yes and your comment helps a ton. I should be able to split up the integral, and then one of the terms is going to be equal to zero as you said (since that one does not include singularities) and then the only thing that is going to be left is the term of the simple-closed curve. But this leads me to wonder why the author did not include the "stronger" version of the formula by rewriting it for loops since none of the closed curves that do not include the singularity should matter. $\endgroup$ Commented Jan 19, 2022 at 21:58
  • $\begingroup$ Maybe the author of the text you read did not include tha version of the theorem because they didn't consider it as a "stronger" version of the theorem. Not only because what we've already seen, but because the applications of contour integration along loops is very rare. Most books and lectures are focused in simple-closed contours or boundaries of multiply-conected regionss. Just my curiosity, which applications were you looking for those loops? $\endgroup$
    – Luciano
    Commented Jan 19, 2022 at 22:07
  • $\begingroup$ I just found it weird that the author two pages before talked about Deformation of Contours and Cauchy's Integral Theorem by using loops and not simple-closed contours. But it is probably as you said and just a choice the author made. I'm trying to create a video that goes over all the methods used in complex analysis for integration and create a flowchart that can be used to pick the right method depending on the integral. I wrote down that Cauchy's Theorem works for loops and therefore I would like to make sure that Cauchy's Integral Formula also works for loops to simplify the flowchart. $\endgroup$ Commented Jan 19, 2022 at 22:14

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